It was conjectured that
one-dimensional quasi-periodic Schrödinger operator generically has Cantor spectrum [1]}. For partial advances we refer to [2]}.
Then people are also curious about how large are the gaps. There are many works along this line: For analytic small potentials, Moser-Pöschel [3]} proved that \( G_k(v) \)  is exponentially small with respect to \( |k| \)  provided that \( |k| \)  large enough and \( \langle k,\alpha \rangle \)  not too close to other \( \langle m,\alpha \rangle \) ; Amor [4]} proved the all gaps \( G_k(v) \)  are sub-exponentially small with respect to \( |k| \)  for small analytic potentials and Diophantine \(\alpha \) ; Damanik-Goldstein [5]} proved the decay rate \( |G_k(v)|\le 2\Vert v\Vert _{r_0}e^{-\pi r_0|k|} \)  for sufficiently small \( v\in C^\omega _{r_0}(d,{\mathbb {R}}) \) . Later on, Leguil-You-Zhao-Zhou [6]} improved the decay rate as \( |G_k(v)|\le \Vert v\Vert _{r_0}^{\frac{2}{3}}e^{-2\pi r|k|} \) , where \( r\in (0,r_0) \)  can be arbitrary close to \( r_0 \) . For weakly coupled quasi-periodic Schrödinger operators with partial Liouville frequencies, see also the result by Liu-Shi [7]}, the size of the spectral gaps decays exponentially.
